When you feel you have read enough examples, you might want to try
the following very simple example on your own. First define the
notion of the ``fringe'' of a tree, where we identify trees simply
as cons structures, with atoms at the leaves. For
example:

ACL2 !>(fringe '((a . b) c . d))
(A B C D)

Next, define the notion of a ``leaf'' of a tree, i.e., a predicate
leaf-p that is true of an atom if and only if that atom appears
at the tip of the tree. Define this notion without referencing the
function fringe. Finally, prove the following theorem, whose
proof may well be automatic (i.e., not require any lemmas).